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On the global attractivity for a logistic equation with piecewise constant arguments. (English) Zbl 1050.34116
The authors consider the following logistic equation with piecewise constant arguments $$ \align\frac{dN(t)}{dt}&= rN(t)\left\{1-\sum_{j=0}^ma_jN([t-j])\right\},\quad t\ge0,m\ge1,\\ N(0)&=N_0>0,\quad N(-j)=N_{-j}\ge0,\quad j=1,2,\dots,m,\endalign $$ where $r>0$, $a_0,a_1,\dots,a_m\ge0$, $\sum_{j=0}^ma_j>0$, and $[x]$ means the maximal integer not greater than $x$. The sequence $\{N_n\}_{n=0}^\infty$, where $N_n=N(n), n=0,1,2,\dots$, satisfies the difference equation $$ N_{n+1}=N_n\exp\left\{r\left(1-\sum_{j=0}^ma_jN_{n-j}\right)\right\},\quad n=0,1,2,\dots\ . $$ Under the condition that the first term $a_0$ dominates the other $m$ coefficients $a_i, 1\le i\le m$, the authors establish new sufficient conditions for the global asymptotic stability of the positive equilibrium $N^*=1/(\sum_{j=0}^ma_j)$.

34K20Stability theory of functional-differential equations
Full Text: DOI
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